Multi-Echelon Supply Chains with Lead Times and Uncertain Demands

Abstract

The aim of this work is to formulate a multi-echelon supply chain and schedule the productions, transportations, and inventories in this supply chain. This supply chain consists of a production echelon, a warehouse echelon, and a retailer echelon. Each echelon consists of multiple nodes. There are challenges of having lead times and uncertain demands in formulating and solving such a supply chain, mathematically. The lead times are incorporated in the formulation of this problem. To overcome the uncertainty of demand, two solution approaches are devised. Each solution approach utilizes two methods to obtain solutions. Solution methods are compared against Cplex solvers and their advantages are explored, computationally. To compare solution approaches, two external measures are used, which are called excess inventory and backlogging. The numerical analyses show a trade-off between solution approaches regarding these measures. In another word, it is shown that one-solution approach results in a lower backlogging, while the other results in a lower excess inventory. These solution approaches and methods provide options for decision makers, in which they can select them considering the priorities and needs.

Appendix 1

1.1 List of Sets, Parameters, and Variables

Sets and indices
T	: the set of time periods $T=\{1,2,\ldots ,|T|\}$
t, r	: indices used for periods, $t,r\in T$
I	: the set of items $I=\{1,2,\ldots ,|I|\}$
i	: index used for items, $i\in I$
$V^p$	: the set of production nodes $V^p=\{1,...,|V^p|\}$
$V^d$	: the set of warehouses/distribution nodes $V^d=\{1,...,|V^d|\}$
$V^r$	: the set of retailer nodes $V^r=\{1,...,|V^r|\}$
V	: the set of all nodes $V=V^p\cup V^d\cup V^r$
u, w	: indices used for network nodes
Parameters
$c_{iu}^t$	: production cost of item i in period t at node $u\in V^p$
$f_{iuw}^t$	: transportation cost of one unit of item i from node u to node w in period t
$h_{iu}^t$	: inventory cost of item i in period t at unit u
$s_{iu}^t$	: setup cost of producing item i in period t at unit $u\in V^p$
$o_{iu}^t$	: startup cost of producing item i in period t at unit $u\in V^p$
$lb_{D_{iu}^{t}}$	: lowest possible demand of item i in period t at unit $u\in V^r$
$ub_{D_{iu}^{t}}$	: highest possible demand of item i in period t at unit $u\in V^r$
$r_{u}$	: production capacity of unit $u\in V^r$
$t_{uw}$	: lead time between nodes u and w (mostly travel time)
$M^t$	: a big number usually set to be equal to the demand in period $t\in T$
Variables
$P_{iu}^t$	: production quantity of item i in period t at node $u\in V^p$
$X_{iuw}^{t}$	: transportation quantity of item i in period t from node u to node w
$Y_{iu}^{t}$	: $Y_{iu}^{t}=1$ if we produce i in period t at node $u\in V^p$, $y_{i}^{t}=0$ otherwise
$Q_{iu}^{t}$	: inventory quantity of item i in period t at unit $u\in V^p$
$D_{iu}^{t}$	: the realization of demand of item i in period t at unit $u\in V^r$
$Z_{iu}^{t}$	: it is one if we produce i in period t at node $u\in V^p$ and not in period $t-1$, zero otherwise.